Tips and Tricks: Logarithms

# Tips and Tricks: Logarithms | Quantitative Aptitude for SSC CGL PDF Download

## Definition

A logarithm serves as the inverse function of exponentiation. It represents the power to which a specific number must be raised to yield another given number. Introduced by John Napier in the early 17th century, logarithms aimed to streamline calculations.

### Logarithms are categorized into two types

• Common Logarithm: Logarithms with a base of 10 are termed common logarithms.
• Natural Logarithm: Logarithms with a base of 'e' are known as natural logarithms.

### Example of Logarithm:

Example: log 100 = 2
Taking LHS
log 102
= 2 log 10
= 2 × 1     (∴ Log 10 = 1)
= 2.

Let’s have a look on some question that will help better in understanding Logarithm.

Basic Difference Between Logarithm and Natural Logarithm:

• Logarithm: Picture having a number and wondering how many instances you must multiply a designated "base" number to reach that given number. The response to this query is the logarithm of the specified number.
• Natural Logarithm: This is a distinctive kind of logarithm that employs a particular base denoted as "e" (a unique number approximately equal to 2.71828). It indicates the number of times you must multiply "e" to achieve a specific number.

### Logarithm Tips, Tricks and Shortcuts

Q1: Which of the following statement is not correct?
(a) log (1 + 2 + 3) = log 1 + log 2 + log 3
(b) log( 2+3) = log (2×3)
(c) log10 1 = 0
(d) log10 10 = 1
Ans: (b)
(a)  log( 1+2+3) = log6 = log (1×2×3) = log1+ log2+log3
(b) log (2 + 3) = log 5 and log (2 x 3) = log 6 = log 2 + log 3
log (2 + 3) ≠ log (2 x 3)
(c) Since,  loga 1=0,
so log10 1=0
(d) Since, loga a = 1
so, log10 10 = 1.

Q3: Solve for x: log2 (x + 3) + log2 (x – 1) = 3
Sol:
Combine the logarithms using the logarithm property log(a) + log(b) = log(a * b):
log2 ((x + 3)×(x − 1)) = 3
23 =(x+3)×(x−1)
8 = (x + 3) × (x − 1)
8 = x2 + 2x − 3
x2 + 2x − 11 = 0
(x + 4)(x - 2) = 0
x = -4, 2

Q2: If X is an integer then solve(log2 X)2– log2x4-32 = 0
Sol: Let us consider,(log2 X)2– log2x4-32 = 0 —— equation 1
let log2 x= y
equation 1 = y2– 4y-32=0
y2-8y+ 4y – 32= 0
y(y-8) + 4(y-8) =0
(y-8) (y+4)= 0
y=8, y= -4
log2X = 8 or, log2X = -4
X = 28 = 256, 0r
Since, X is an integer so X = 256.

The document Tips and Tricks: Logarithms | Quantitative Aptitude for SSC CGL is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
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## FAQs on Tips and Tricks: Logarithms - Quantitative Aptitude for SSC CGL

 1. What is a logarithm?
Ans. A logarithm is the inverse operation of exponentiation. It is a mathematical function that calculates the power to which a base number must be raised to obtain a given number. In simpler terms, a logarithm helps us find the exponent or power that produces a specific result when multiplied by a base number.
 2. Why are logarithms useful in mathematics?
Ans. Logarithms are widely used in mathematics because they simplify complex calculations involving exponential growth or decay. They help condense large numbers into more manageable forms and facilitate solving equations involving exponential functions. Logarithms also have applications in various fields such as physics, engineering, finance, and computer science.
 3. What are the properties of logarithms?
Ans. Logarithms have several important properties that make them useful in solving mathematical problems. Some of the key properties include the product rule (log(ab) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^n) = n * log(a)). These properties allow for simplification and manipulation of logarithmic expressions.
 4. How do logarithms help solve exponential equations?
Ans. Logarithms help solve exponential equations by allowing us to isolate the variable in the exponent. By taking the logarithm of both sides of the equation, we can bring the exponent down as a coefficient, making it easier to solve for the variable. This process is especially useful when the variable appears in both the base and exponent of an exponential equation.
 5. Can logarithms be negative or zero?
Ans. Logarithms can only be calculated for positive numbers. The logarithm of zero is undefined, as there is no exponent that can raise the base to zero to obtain zero. Similarly, the logarithm of a negative number is also undefined in the real number system. However, logarithms of negative numbers can be defined in complex number systems.

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